Let `S_1, S_2 and S_3` be the sum of n terms of 3 arithmetic series, the first termof each being 1 and the respective common differences are 1,2,3,then prove that `S_1+S_2+2S_2`.

Explanation

` S_(1) = (n)/(2) [ 2xx 1 + (n-1) xx1] `
` = (n)/(2) (n+1)` …(i)
` S_(2) = (n)/(2) [ 2xx 1 + (n-1) xx2] = (n)/(2) (2n) = n^(2)` …(ii)
` S_(3) = (n)/(2) [ 2xx 1 + (n-1) xx 3 ] = (n)/(2) (3n -1) ` …(iii)
Adding (i) and (iii) , we get
`S_(1) + S_(3) = (n)/(2) (n + 1 + 3n - 1) = (n)/(2) (4n) = 2 xx n^(2) = 2S_(2)`
` therefore S_(1) + S_(3) = 2S_(2)` .